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29
A graph cut algorithm for higherorder markov random fields
 IN: INT. CONF. COMPUTER VISION
, 2011
"... Higherorder Markov Random Fields, which can capture important properties of natural images, have become increasingly important in computer vision. While graph cuts work well for firstorder MRF’s, until recently they have rarely been effective for higherorder MRF’s. Ishikawa’s graph cut technique ..."
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Higherorder Markov Random Fields, which can capture important properties of natural images, have become increasingly important in computer vision. While graph cuts work well for firstorder MRF’s, until recently they have rarely been effective for higherorder MRF’s. Ishikawa’s graph cut technique [8, 9] shows great promise for many higherorder MRF’s. His method transforms an arbitrary higherorder MRF with binary labels into a firstorder one with the same minima. If all the terms are submodular the exact solution can be easily found; otherwise, pseudoboolean optimization techniques can produce an optimal labeling for a subset of the variables. We present a new transformation with better performance than [8, 9], both theoretically and experimentally. While [8, 9] transforms each higherorder term independently, we transform a group of terms at once. For n binary variables, each of which appears in terms with k other variables, at worst we produce n nonsubmodular terms, while [8, 9] produces O(nk). We identify a local completeness property that makes our method perform even better, and show that under certain assumptions several important vision problems (including common variants of fusion moves) have this property. Running on the same field of experts dataset used in [8, 9] we optimally label significantly more variables (96 % versus 80%) and converge more rapidly to a lower energy. Preliminary experiments suggest that some other higherorder MRF’s used in stereo [20] and segmentation [1] are also locally complete and would thus benefit from our work.
Markov Random Field Modeling, Inference & Learning in Computer Vision & Image Understanding: A Survey
, 2013
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Generalized Roof Duality for PseudoBoolean Optimization
"... The number of applications in computer vision that model higherorder interactions has exploded over the last few years. The standard technique for solving such problems is to reduce the higherorder objective function to a quadratic pseudoboolean function, and then use roof duality for obtaining a ..."
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Cited by 7 (1 self)
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The number of applications in computer vision that model higherorder interactions has exploded over the last few years. The standard technique for solving such problems is to reduce the higherorder objective function to a quadratic pseudoboolean function, and then use roof duality for obtaining a lower bound. Roof duality works by constructing the tightest possible lowerbounding submodular function, and instead of optimizing the original objective function, the relaxation is minimized. We generalize this idea to polynomials of higher degree, where quadratic roof duality appears as a special case. Optimal relaxations are defined to be the ones that give the maximum lower bound. We demonstrate that important properties such as persistency still hold and how the relaxations can be efficiently constructed for general cubic and quartic pseudoboolean functions. From a practical point of view, we show that our relaxations perform better than stateoftheart for a wide range of problems, both in terms of lower bounds and in the number of assigned variables. 1.
Landmark/Imagebased Deformable Registration of Gene Expression Data
"... Analysis of gene expression patterns in brain images obtained from highthroughput in situ hybridization requires accurate and consistent annotations of anatomical regions/subregions. Such annotations are obtained by mapping an anatomical atlas onto the gene expression images through intensity and/ ..."
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Cited by 6 (4 self)
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Analysis of gene expression patterns in brain images obtained from highthroughput in situ hybridization requires accurate and consistent annotations of anatomical regions/subregions. Such annotations are obtained by mapping an anatomical atlas onto the gene expression images through intensity and/or landmarkbased registration methods or deformable modelbased segmentation methods. Due to the complex appearance of the gene expression images, these approaches require a preprocessing step to determine landmark correspondences in order to incorporate landmarkbased geometric constraints. In this paper, we propose a novel method for landmarkconstrained, intensitybased registration without determining landmark correspondences a priori. The proposed method performs dense image registration and identifies the landmark correspondences, simultaneously, using a single higherorder Markov Random Field model. In addition, a machine learning technique is used to improve the discriminating properties of local descriptors for landmark matching by projecting them in a Hamming space of lower dimension. We qualitatively show that our method achieves promising results and also compares well, quantitatively, with the expert’s annotations, outperforming previous methods. 1.
Generalized Roof Duality for MultiLabel Optimization: Optimal
"... Abstract. We extend the concept of generalized roof duality from pseudoboolean functions to realvalued functions over multilabel variables. In particular, we prove that an analogue of the persistency property holds for energies of any order with any number of linearly ordered labels. Moreover, we ..."
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Abstract. We extend the concept of generalized roof duality from pseudoboolean functions to realvalued functions over multilabel variables. In particular, we prove that an analogue of the persistency property holds for energies of any order with any number of linearly ordered labels. Moreover, we show how the optimal submodular relaxation can be constructed in the firstorder case.
Maximum Persistency in Energy Minimization
"... We consider discrete pairwise energy minimization problem (weighted constraint satisfaction, maxsum labeling) and methods that identify a globally optimal partial assignment of variables. When finding a complete optimal assignment is intractable, determining optimal values for a part of variable ..."
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Cited by 4 (1 self)
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We consider discrete pairwise energy minimization problem (weighted constraint satisfaction, maxsum labeling) and methods that identify a globally optimal partial assignment of variables. When finding a complete optimal assignment is intractable, determining optimal values for a part of variables is an interesting possibility. Existing methods are based on different sufficient conditions. We propose a new sufficient condition for partial optimality which is: (1) verifiable in polynomial time (2) invariant to reparametrization of the problem and permutation of labels and (3) includes many existing sufficient conditions as special cases. We pose the problem of finding the maximum optimal partial assignment identifiable by the new sufficient condition. A polynomial method is proposed which is guaranteed to
Partial optimality by pruning for MAPinference with general graphical models
 In CVPR
, 2014
"... We consider the energy minimization problem for undirected graphical models, also known as MAPinference problem for Markov random fields which is NPhard in general. We propose a novel polynomial time algorithm to obtain a part of its optimal nonrelaxed integral solution. Our algorithm is initiali ..."
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We consider the energy minimization problem for undirected graphical models, also known as MAPinference problem for Markov random fields which is NPhard in general. We propose a novel polynomial time algorithm to obtain a part of its optimal nonrelaxed integral solution. Our algorithm is initialized with variables taking integral values in the solution of a convex relaxation of the MAPinference problem and iteratively prunes those, which do not satisfy our criterion for partial optimality. We show that our pruning strategy is in a certain sense theoretically optimal. Also empirically our method outperforms previous approaches in terms of the number of persistently labelled variables. The method is very general, as it is applicable to models with arbitrary factors of an arbitrary order and can employ any solver for the considered relaxed problem. Our method’s runtime is determined by the runtime of the convex relaxation solver for the MAPinference problem. 1.
Nonrigid 2D3D Medical Image Registration using Markov Random Fields
, 2013
"... Abstract. The aim of this paper is to propose a novel mapping algorithm between 2D images and a 3D volume seeking simultaneously a linear plane transformation and an inplane dense deformation. We adopt a metric free locally overparametrized graphical model that combines linear and deformable param ..."
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Cited by 3 (1 self)
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Abstract. The aim of this paper is to propose a novel mapping algorithm between 2D images and a 3D volume seeking simultaneously a linear plane transformation and an inplane dense deformation. We adopt a metric free locally overparametrized graphical model that combines linear and deformable parameters within a coupled formulation on a 5dimensional space. Image similarity is encoded in singleton terms, while geometric linear consistency of the solution (common/single plane) and inplane deformations smoothness are modeled in a pairwise term. The robustness of the method and its promising results with respect to the state of the art demonstrate the extreme potential of this approach.
A PrimalDual Algorithm for HigherOrder Multilabel Markov Random Fields
"... Graph cuts method such as αexpansion [4] and fusion moves [22] have been successful at solving many optimization problems in computer vision. Higherorder Markov Random Fields (MRF’s), which are important for numerous applications, have proven to be very difficult, especially for multilabel MRF’s ..."
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Graph cuts method such as αexpansion [4] and fusion moves [22] have been successful at solving many optimization problems in computer vision. Higherorder Markov Random Fields (MRF’s), which are important for numerous applications, have proven to be very difficult, especially for multilabel MRF’s (i.e. more than 2 labels). In this paper we propose a new primaldual energy minimization method for arbitrary higherorder multilabel MRF’s. Primaldual methods provide guaranteed approximation bounds, and can exploit information in the dual variables to improve their efficiency. Our algorithm generalizes the PD3 [19] technique for firstorder MRFs, and relies on a variant of maxflow that can exactly optimize certain higherorder binary MRF’s [14]. We provide approximation bounds similar to PD3 [19], and the method is fast in practice. It can optimize nonsubmodular MRF’s, and additionally can incorporate problemspecific knowledge in the form of fusion proposals. We compare experimentally against the existing approaches that can efficiently handle these difficult energy functions [6, 10, 11]. For higherorder denoising and stereo MRF’s, we produce lower energy while running significantly faster. 1. Higherorder MRFs There is widespread interest in higherorder MRF’s for problems like denoising [23]and stereo [30], yet the resulting energy functions have proven to be very difficult to minimize. The optimization problem for a higherorder MRF is defined over a hypergraph with vertices V and cliques C plus a label set L. We minimize the cost of the labeling f: LV  → < defined by f(x) =
Structured learning of sumofsubmodular higher order energy functions
, 1309
"... Submodular functions can be exactly minimized in polynomial time, and the special case that graph cuts solve with max flow [18] has had significant impact in computer vision [5, 20, 27]. In this paper we address the important class of sumofsubmodular (SoS) functions [2, 17], which can be efficient ..."
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Submodular functions can be exactly minimized in polynomial time, and the special case that graph cuts solve with max flow [18] has had significant impact in computer vision [5, 20, 27]. In this paper we address the important class of sumofsubmodular (SoS) functions [2, 17], which can be efficiently minimized via a variant of max flow called submodular flow [6]. SoS functions can naturally express higher order priors involving, e.g., local image patches; however, it is difficult to fully exploit their expressive power because they have so many parameters. Rather than trying to formulate existing higher order priors as an SoS function, we take a discriminative learning approach, effectively searching the space of SoS functions for a higher order prior that performs well on our training set. We adopt a structural SVM approach [14, 33] and formulate the training problem in terms of quadratic programming; as a result we can efficiently search the space of SoS priors via an extended cuttingplane algorithm. We also show how the stateoftheart max flow method for vision problems [10] can be modified to efficiently solve the submodular flow problem. Experimental comparisons are made against the OpenCV implementation of the GrabCut interactive segmentation technique [27], which uses handtuned parameters instead of machine learning. On a standard dataset [11] our method learns higher order priors with hundreds of parameter values, and produces significantly better segmentations. While our focus is on binary labeling problems, we show that our techniques can be naturally generalized to handle more than two labels. 1.